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Objectives
- To display a published case study using plug-and-play methods with non-trivial model complexities
- To demonstrate the use of profile likelihood in scientific inference
- To discuss the interpretation of parameter estimates
- To emphasize the potential need for extra sources of stochasticity in modeling
Measles revisited
Motivation: challenges in inference from disease dynamics
- Understanding, forecasting, managing epidemiological systems increasingly depends on models.
- Dynamic models can be used to test causal hypotheses.
- Real epidemiological systems:
- are nonlinear
- are stochastic
- are nonstationary
- evolve in continuous time
- have hidden variables
- can be measured only with (large) error
Dynamics of infectious disease outbreaks illustrate this well.
- Measles is the paradigm for a nonlinear ecological system that can be well described by low-dimensional nonlinear dynamics.
- A tradition of careful modeling studies have proposed and found evidence for a number of specific mechanisms, including
- a high value of \(R_0\) (c. 15–20)
- seasonality in transmission rates associated with school terms
- a birth cohort effect
- response to changing birth rates
- under-reporting
- fadeouts and reintroductions that scale with city size
- spatial traveling waves
Much of this evidence has been amassed from fitting models to data, using a variety of methods.

Outline
- We revisit a classic measles data set, weekly case reports in 954 urban centers in England and Wales during the pre-vaccine era (1950–1963).
- We revisit questions of:
- measles extinction and recolonization
- transmission rates
- seasonality
- resupply of susceptibles
- We use a model that
- expresses our current understanding of measles dynamics
- includes a long list of mechanisms that have been proposed and demonstrated in the literature
- cannot be fit by existing likelihood-based methods
- We examine data from large and small towns using the same model, something no existing methods have been able to do.
- We ask: does our perspective on this disease change when we expect the models to explain the data in detail?
- What bigger lessons can we learn regarding inference for dynamical systems?
He, Ionides, & King, J. R. Soc. Interface (2010)
Data sets
- Twenty towns, including
- 10 largest
- 10 smaller, chosen at random
- population sizes: 2k–3.4M
- Weekly case reports, 1950–1963
- Annual birth records and population sizes, 1944–1963

